EFNO让神经算子跨域通用，解决传统方法依赖周期域的问题
Euclidean Fourier Neural Operators
- 用物理波矢连续函数替代离散频域索引，实现域无关建模
- 在热方程和晶体结构任务中均实现未见网格与域的泛化
- 适合需要跨尺度、跨结构迁移的科学计算场景
傅里叶神经算子（FNO）能高效学习函数空间间的映射，因其训练与评估时对网格分辨率具有不变性。然而，FNO对周期性定义域不具不变性：其离散谱权重以整数傅里叶模态编号索引，对应物理波矢。当应用于不同域时，相同训练权重作用于不同波矢，导致隐式表示不同算子，限制了跨域迁移能力。为此，本文提出欧几里得傅里叶神经算子（EFNO），通过将谱核参数化为物理波矢的连续函数，使算子在不同形状与尺寸的周期域上保持一致行为。我们在简单热方程及材料科学中的交换-关联势学习任务（涉及不同晶格结构）上评估了EFNO，结果表明其可有效泛化至未见过的网格大小与域形。该方法为跨域物理系统建模提供了新范式。
原文摘要 · Abstract (English)
Fourier neural operators (FNOs) provide an efficient framework for learning mappings between function spaces as they are, by construction, independent of the grid resolution at which they are trained and evaluated. However, FNOs are not independent of the periodic domain they are applied to: their discrete spectral weights are indexed by integer Fourier mode numbers, which correspond to physical wavevectors. When applied to a different domain, the same trained weights act at different wavevectors, and the FNO silently represents a different operator. This makes FNOs unsuitable for tasks where transfer across domains is crucial. We propose Euclidean Fourier neural operators~(EFNOs) as a domain-independent alternative to FNOs. By parameterizing the spectral kernel as a continuous function of the physical wavevector, the EFNO can learn operators that act consistently across periodic domains of varying shape and size. We evaluate the EFNO on a simple heat equation and on a practically relevant materials science task of learning exchange-correlation potentials across different crystal structures, and demonstrate that the EFNO is able to generalize to unseen grid sizes and domains.
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